What academic level would one need to be at to fully understand papers published on the twin prim conjecture? Specifically, what academic level would one need to be at to fully understand Goldston-Pintz-Yildirim's work on twin primes? Undergraduate, Graduate, or PhD?

This is a rather subjective question, and it depends on _specialization_ : I suspect there are "pre-docs" (even pre-"grads") studying number theory who would better understand the paper than some PhD-level mathematicians, in say, group theory. Mathematical maturity, and area (and level) of specialization is of greater importance than the degree one holds. It also depends how deeply you want to understand any published paper, including the paper you refer to (and what you mean by "fully" understand.)

A good look at the references will help indicate what prior knowledge and/or familiarity with such work is assumed, as will the abstract. References will often include work that may be a bit more general, and you can backtrack to that work, to check and see what level of familiarity is assumed to understand _that_ work.